Searching for Self-Similarity in GPRS
Roger Kalden, Sami Ibrahim
Ericsson Research, Ericsson Eurolab Deutschland GmbH, Aachen, Germany
Roger.Kalden@ericsson.com
, phone +49 2407 575 7831
Abstract. Based on measurements in live GPRS networks, the degree of self-
similarity for the aggregated WAP and WEB traffic is investigated by utilizing
six well established Hurst parameter estimators. We show that in particular
WAP traffic is long-range dependent and its scaling for time scales below the
average page duration is not second order self similar. WAP over UDP can also
determine the overall traffic scaling, if it is the majority traffic. Finally we ob-
serve that the minor traffic exhibits a larger Hurst value than the aggregated
traffic, in case of WAP as well as in case of WEB traffic.
1. Introduction
Based on live GPRS traffic measurements, we investigate the packet arrival proc-
ess and the data volume arrival process of WAP and WEB traffic on statistical self-
similarity.
Many studies have been looking at various network types and found evidence for
self-similarity (e.g., [1],[2],[3]). This property is regarded as an invariant of network
traffic and has serious performance implications. In the case of self-similar traffic the
applied statistics for performance analysis and network dimensioning are different
from those when applied to statistically more simple traffic, which can be modeled
with Markovian processes ([4],[5]). For instance the queue tail behavior is heavy-
tailed in the case of self-similar input traffic [6]. This leads to heavy-tailed packet de-
lay distributions, which can influence the TCP round-trip time estimations. Further-
more, the traffic does not smooth out in the case of aggregation, leading to congestion
situations and packet-drops due to the burstiness of the traffic. Consequently, it is im-
portant to understand the self-similar nature of the traffic in a network in order to ap-
ply the right statistical methods for performance investigations and network dimen-
sioning. In previous studies the reason for self-similarity has been identified as the
heavy-tailedness of many statistical properties of Internet traffic, on the user and ap-
plication level. In [7] and [18] the authors showed that heavy-tailed sessions and file
size lengths lead to self-similarity for large aggregation scales (long-range depend-
ency). The authors in [8] and [14] have furthermore shown that self-similarity (in par-
ticular the multiscaling) behavior at small timescales (e.g., smaller than the average
round-trip time) is due to protocol interactions of TCP. The biggest sale of poker tables and poker chips for 3 days only.
But GPRS traffic and in general cellular access network traffic has not yet been in-
vestigated on its self-similarity property. GPRS is not merely a new access technol-
ogy, it also introduces novel applications such as WAP and MMS, and provides Inter-
net access in a mobile or nomadic usage environment. This yields a special traffic
composition, different from wireline Internet traffic. We have investigated GPRS net-
works with more than 60% of the traffic volume consisting of UDP traffic. This is in
sharp contrast to the usual 80% of TCP traffic in the fixed Internet (cf. [9] for earlier
figures on GPRS traffic). Additionally, WAP and MMS file sizes are in general much
shorter than WEB and FTP file sizes [17]. For these reasons we counter-check the
property of self-similarity in GPRS. Based on our results, which show that GPRS and
in particular WAP traffic, is asymptotically second order self-similar (long-range de-
pendent), we propose for further research to study the scaling nature of GPRS traffic
and to explore in particular the reasons for this for WAP traffic.
The remainder of the paper is structured as follows. Firstly, we give a brief over-
view of self-similarity together with the commonly used methods to test self-
similarity. Next, we describe our measurement set-up and the investigated traces. In
our main section we present the results of the Hurst parameter estimation methods to
assess the degree of self-similarity in GPRS. Finally, in the concluding section we list
open issues and future work items.
2. Self-Similarity
Self-similarity, in a strict sense, means that the statistical properties (e.g., all mo-
ments) of a stochastic process do not change for all aggregation levels of the stochas-
tic process. That is, the stochastic process "looks the same" if one zooms in time "in
and out" in the process. LetAll poker chip sets prices have been reduced by 25 percent.
...)
,
2
,
1
,
0
:
(
=
=
t
X
X
t
be a covariance-stationary stochas-
tic process, with constant and finite mean and finite variance and autocorrelation
function
)
(k
, which only depends on k. Further, let
...)
,
3
,
2
,
1
:
(
)
(
)
(
=
=
k
X
X
m
k
m
be a
new time series generated by averaging the original time series
X
over non-
overlapping blocks of size m. That is, for each
...
,
3
,
2
,
1
=
m
the series is given by
...
,
3
,
2
,
1
),
...
(
/
1
1
)
(
=
+
+
=
+
-
k
X
X
m
X
km
m
km
m
k
and
)
(
)
(
k
m
the corresponding autocor-
relation function. The stochastic process
...)
,
2
,
1
,
0
:
(
=
=
t
X
X
t
is then called exactly
second-order self-similar with self-similarity parameter
2
/
1 ß
H
-
=
, if the autocorre-
lation functions
)
(
)
(
k
m
of the processes
)
(m
X
are all equal to the autocorrelation
function
)
(k
of the original process
X
. That is:
...
,
3
,
2
,
1
),
(
)
(
)
(
=
=
k
k
k
m
for all
...
,
3
,
2
,
1
=
m
X
is called asymptotically second-order self-similar with self-
similarity parameter
2
/
1 ß
H
-
=
if the correlation structure of the aggregated time
series
)
(m
X
is indistinguishable from the correlation structure of the original time se-
ries
X
as
m
. That is, the above holds asymptotically for large aggregation lev-
els.
H expresses the degree of self-similarity; large values indicate stronger self-
similarity. If H
(0.5,1) X is called long-range dependent (LRD). Both, exactly and
asymptotically second-order self-similar traffic can include long-range dependency,
however, as it is often the used case, we will use long-range dependent for asymptoti-
cally second-order self-similar traffic. medi fast
Asymptotical second-order self-similar processes have a few interesting character-
istics. For instance, the autocorrelation function of an LRD process is decaying hy-
perbolically. That is (with
the same as in H, above):
( )
-
k
k
k
~
lim
(0 <
< 1).
Also the variance of the aggregated time series is very slowly decaying. That is:
_
( )
[ ]
-
m
X
Var
m
~
(0 <
< 1).
Another property is that the power spectrum is singular as the frequency is ap-
proaching 0. That is:
)
1
(
1
~
)
(
-
w
w
telefon dinleme                    
S
w
(0 <
< 1).
3. Estimation Methods for Self-Similarity
Various methods for estimating the Hurst parameter H exist for deducing self-
similarity or long-range dependency [15]. The estimation methods can be grouped
into time-based and frequency-based methods. We will briefly provide an overview of
the methods used to estimate the value of the Hurst parameter.

The first four methods are time-based:
R/S method
This method is based on empirical observations by Hurst and estimates H are based
on the R/S statistic. It indicates (asymptotically) second-order self-similarity. H is
roughly estimated through the slope of the linear line in a log-log plot, depicting the
R/S statistics over the number of points of the aggregated series.
Variance Method
The Variance Method is based on the slowly decaying variance property as stated
above. It indicates long-range dependency. The slope
of the straight line in a log-log
plot, depicting the sample variance over the block size of each aggregation, is used for
roughly estimating H. H is given by
2
1
-
=
H
.
Absolute Moment Method
This method is related to the variance method computed for the first moment. The
slope
of the straight line in a log-log plot, depicting the first moment of the aggre-
gated block over the block size, provides an estimator for H, by H = 1+
.
Ratio of Variance of Residuals
This uses the empirical observation, that is, the sample variance of residuals, plot-
ted over the aggregation level, yields a slope equivalent to roughly 2H. It indicates
some self-similarity.

The next two methods are frequency-based:
Periodogram method
This method is based on the power-spectrum singularity at 0-property as stated
above. The slope of the straight line, approximating the logarithm of the spectral den-
sity over the frequency as the frequency approaches 0, yields H.
Abry-Veitch method
This method is based on the multi-resolution analysis and the discrete wavelet
transformation. H is estimated by fitting a straight line to the energy in the series over
octave j (expressing the scaling level in the time and the frequency domain) in a log-
log plot.
1
This method is the most comprehensive and robust method for determining
the scaling behavior of traffic traces. It strength follows from the fact that the multi-
resolution analysis itself has a structural affinity to the scaling process under study.
That is, multi-resolution analysis itself exploits scaling, but transfers the complex
scaling process to a much simpler wavelet domain, in which short range dependent
(SRD) statistics can be applied to infer answers on the scaling of the process [10].
We show results from our traces for all mentioned methods. In particular we will
use the Abry-Veitch method to derive the scaling nature of the process.
All used methods provide some intermediate statistics, which is used to derive the
Hurst value. For instance, in the case of the Variance Method, these are the aggre-
gated variance values for each aggregation level; or, in the case of the Abry-Veitch
method, the intermediate statistics used are the wavelet coefficients. Based on those
values linear regression is used to fit a straight line to derive the Hurst value.
Important to consider is that typically the linear regression should not consider all
of the values from the intermediate statistic. In case of the R/S method, the Variance
Method and the Absolute Moment method, it is recommended not to use the results of
the first few aggregations levels and neither the last few aggregation levels. The rea-
son for this is that these values are not very reliable because either the aggregation
level is too low (sampling too few points per block) or it is too high (sampling all
points in just a few blocks). In the case of the Periodogram Method it is recommended
only to use approximately the first 10 percent of the results, close to the frequency 0.
This is justified by the asymptotic LRD property close to the frequency 0. The Abry-
Veitch method is the most robust of all estimation methods [10]. It actually shows the
scaling of the process over all aggregation levels (octave j), which allows to optimally
select an appropriate starting point for the regression. This starting point is indicated
by a
2
-goodness-of-fit test. In the case of assumed LRD traffic, the regression line is
fitted from this starting point to the largest available octave in the data.
We use the SELFIS tool [11] to derive the intermediate statistics for all but the
Abry-Veitch method. Additionally, we use the LDestimate-code which implements
the Abry-Veitch method [12]. In the case of the SELFIS tool we applied our own lin-
ear regression on the intermediate results obtained from SELFIS, as explained above.
This is necessary as SELFIS estimates H, always based on a linear regression over all
available points, which can heavily bias the results due to outliers at the end points.
For this reason we shortened the intermediate statistical results to all but the first 2
and the last 2 aggregation levels. In all cases, applying the manual regression, the fit
is better than the SELFIS tool directly provides. The differences in the values of H,
between manually applied linear regression and the final results of SELFIS, are some-
1
More precisely, based on the wavelet coefficient d
j,k
the amount of energy |d
j,k
|
2
in the signal at
about the time t=2
j
k and about the frequency 2
-j
is measured. [ |d
j,k
|
2
] is the amount of en-
ergy at octave j. If the initial resolution of the time series is t
0
(bin size), the time resolution at
each scaling level j is t
j
=2
j
t
0
.
times quite large, which stresses the importance to apply manual post-processing. The
LD estimator function for the Abry-Veitch test suggests an optimal starting point for j
which we always used. It furthermore plots the scaling behavior over all octaves j, al-
lowing to judge on the type of scaling. We discuss those results as well.
4. Data
Traces
In cooperation with Vodafone we conducted measurements in two live GPRS net-
works. We captured the IP packet headers at the Gi interface for all users in a geo-
graphical region of the operator's GPRS network. The Gi interface connects the mo-
bile network to external packet switched networks (like the Internet, a corporate
Intranet or Email and WAP/MMS proxies). All IP packets from or to mobile termi-
nals traverse this interface. The traces we focus on were taken during summer 2003.
We collect for every packet crossing the network monitor interface a time stamp,
with 1µs accuracy, with the total length of the packet in bytes. For our further investi-
gations we generate from the original trace the Packet Arrival Process (PAP) as a dis-
crete time-series process by counting the number of packets and the Data Volume
Process (DVP) as the total number of bytes within a time interval (bin) or 100 ms.
We are interested in the scaling behavior for the aggregated traffic, WAP oriented
traffic and WEB oriented traffic. For this purpose we look at three "sup-"sampled
traces. Firstly, we investigate the total aggregated traffic (up and downlink traffic),
which we measured on the Gi interface. Next, we have split-up the traffic into WEB
oriented traffic and WAP oriented traffic. We do this by splitting the data according to
the APN (Access Point Names) they are belonging to. GPRS allows users to attach to
different APNs provided by the operator. Typically, the APNs are used for different
types of traffic and split logically the traffic on the Gi interface. We checked our
measurements for the used applications and found that most of the traffic on one APN
consists of WEB like applications, including HTTP, FTP, Email, etc. On the other
APN we see mostly WAP like traffic, comprising WAP and MMS.
2
However, the
APNs do not filter for applications, hence it occasionally happens that we see WEB
traffic on the WAP APN and vice versa.
On one of the network measurement points (Vfe1) the traffic splits up into 25%
WEB-APN and 75% WAP APN traffic, while in the other network (Vfe2) the split is
70% WEB APN and 30% WAP APN traffic.
We traced several 24-hour periods and investigated appropriate busy-hour periods
spanning several hours, for each of the networks. Our results will be presented only
for one selected day for each network. In the case of Vfe1 we chose a 110-minute
busy-hour period in the afternoon, in the case of Vfe2 we chose a 430-min (7-hour )
busy-hour period covering the whole afternoon. Since all estimator methods (with ex-
ception of the Abry-Veitch method) require stationarity and often are very sensitive to
underlying trends or correlations in the traffic process [13], we investigated the cho-
sen busy-hour periods on trends by plotting the moving average, on periodicity by in-
vestigating the Fourier transformation, and on stationarity by applying the average run
2
We used the same investigation method we have used in [9] for differentiating the applica-
tions.
test. All tests indicated the suitability of the chosen periods for the estimation meth-
ods, i.e., they appear to be stationary and they have no visible periodicities or trends.
5. Results on Self-Similarity
We present for all processes detailed results obtained by the Abry-Veitch method,
together with Hurst estimations acquired by the other described methods. Table 1 and
Table 2 show the results for the estimated Hurst values by the Abry-Veitch method
for the packet arrival process. The second row `H' contains the Hurst values and in
the third row marked `conf.' the confidence values for the estimated Hurst values are
listed. Figure 1 and Figure 2 illustrate the Hurst values obtained from the different
methods, for comparison. All values are very similar with Hurst values of about 0.8
and higher. This strongly suggests long range dependency for the PAP for both net-
works Vfe1 and Vfe2.
Table 1. Results for PAP in Vfe1
Vfe1 Agg WEB WAP
H 0.86
0.83
1.06
Conf. [0.76,0.95] [0.74,0.92] [0.96,1.14]
Scaling
Fig. 8
Fig. 8
Fig. 7
Table 2. Results for PAP in Vfe2
Vfe2 Agg WEB
WAP
H 0.90
1.02
0.89
Conf. [0.81,0.98]
[0.93,1.11]
[0.79,0.97]
Scaling
Fig. 5
Fig. 5
Fig. 5
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.10
1.20
Hvar
Hrs
Ham
Hrvor
Hper
Ha-v
aggregated
WEB
WAP
Fig. 1. All Hurst values for PAP and Vfe1
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.10
1.20
Hvar
Hrs
Ham
Hrvor
Hper
Ha-v
aggregated
WEB
WAP
Fig. 2. All Hurst values for PAP and Vfe2
`Hvar' stands for Variance Method, `Hrs' for R/S Method, `Ham' for Absolute Moment
Method, `Hrvor' for Variance of Residuals, `Hper' for Periodogram Method and `Ha-v'
for Abry-Veitch Method
In some cases the Hurst value is above 1, actually precluding LRD, but in the case
of the Abry-Veitch method all confidence intervals include also a value of below 1.
Furthermore, as explained next, inspection of the output-plots suggests in such cases
still asymptotic self-similarity.
For the DVP, Table 3 and Figure 3 list the Hurst values for Vfe1 and Table 3 and
Figure 4 list the Hurst values for Vfe2. Again all results strongly indicate LRD.
All estimation methods, except for the Abry-Veitch method, assume an LRD
model beforehand. That is, the Hurst estimation value can only be regarded as correct
if the assumption of an LRD process holds. In contrast, the Abry-Veitch test is not
based on such assumptions. It shows the scaling of the process for all time scales in
the diagram. Only by interpreting the results in this Logscale Diagram, the true nature
of the process is determined (e.g., self-similarity, long-range dependency, multiscal-
ing) [10]. For our traces we have encountered four basic log-scale diagram types, de-
picted in Figure 5 to Figure 8. In Table 1 to Table 4 we list in the 4
th
row for each
process the plot that comes closest, respectively. The individual plots looked very
similar to the exemplary plots, but with different scales on the y-axes.
Table 3. Results of DVP for Vfe1
Vfe1 Agg WEB
WAP
H
0.69 0.68 0.92
Conf.
[0.65,0.72] [0.64,0.71] [0.88,0.96]
Scaling
Fig. 6
Fig. 6
Fig. 7
Table 4. Results of DVP for Vfe2
Vfe2 Agg WEB
WAP
H 0.82
1.07
0.81
Conf. [0.73,0.90]
[0.98,1.15]
[0.72,0.89]
Scaling
Fig 5
Fig. 7
Fig. 5
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.10
1.20
Hvar
Hrs
Ham
Hrvor
Hper
Ha-v
aggregated
WEB
WAP
Fig. 3. All Hurst values for DVP and Vfe1
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1.10
1.20
Hvar
Hrs
Ham
Hrvor
Hper
Ha-v
aggregated
WEB
WAP
Fig. 4. All Hurst values for DVP and Vfe2
`Hvar' stands for Variance Method, `Hrs' for R/S Method, `Ham' for Absolute Moment
Method, `Hrvor' for Variance of Residuals, `Hper' for Periodogram Method and `Ha-v'
for Abry-Veitch Method

Figure 5 and Figure 7 both show a typical plot for LRD traffic. The second-order
scaling starts from a certain scaling point on and continues until the largest available
scale in the trace. For small scales we do not see second-order scaling behavior. We
have found this scaling behavior for all WAP processes.
Figure 6 also exhibits LRD scaling, but actually has two scaling regions. One from
approximately 1 to 8 and one from 8 to maximum scale. This is called bi-scaling. We
have found this in the case of DVP for Web traffic in Vfe2. Figure 8 depicts the case
where the process has second-order scaling over all scales. This indicates strictly sec-
ond-order self-similarity. We see this in the case of PAP for WEB traffic also in Vfe2.
We point out some interesting observations (cf. Table 1 to Table 4). Firstly, the
Hurst value of the aggregated traffic is always very close to the Hurst value of the ma-
jority of the traffic. This is in agreement with [14]. In the case of Vfe1 the major part
of the traffic is WAP traffic, in the case of Vfe2 it is WEB traffic. More in detail, even
the whole scaling behavior, as depicted by the Logscale Diagram, is very similar be-
tween the majority traffic and the aggregated traffic. This implies, by knowing the
scaling of the majority traffic, one obtains also the scaling of the aggregated traffic.
Secondly, the minor traffic has always slightly higher Hurst values, with changing
roles of WAP and WEB in the cases of Vfe1 and Vfe2. We do not have an explana-
tion for this, yet. One reason might be that even so we applied the estimation methods
on separated traces per APN, it is not possible to separate them truly: they have been
both traveling together through the GPRS network, thereby most likely affecting each
other.
2
4
6
8
10
12
22
23
24
25
26
27
28
29
30
31
32
Octave j
y
j
Fig. 5. Typical plot for processes showing
long range dependency. Below a certain
scale no regular linear scaling exists. The
linear part is at j=8 divided in two scaling
regions
2
4
6
8
10
12
14
25
26
27
28
29
30
31
32
33
34
35
Octave j
y
j
Fig. 6. Typical plot for processes showing bi-
scaling. The second scaling region starts at
j=8. It also implies long range dependence
An interesting question arises whether it is actually possible to truly identify the
Hurst value for each type of traffic separately. In [14] the authors have shown that
non-self-similar UDP traffic is affected by self-similar TCP traffic. But this effect is
only strong if the self-similar traffic is the major traffic. In our case we observe high
Hurst values even in the case the WAP (UDP) traffic is dominating. This suggests that
WAP traffic itself is strongly long-range dependent.
Furthermore, we see that WAP traffic has a very different scaling behavior for
small scales compared to WEB traffic. The reason for the different small scaling be-
havior can be assumed to be the very different transport mechanism of TCP and WAP
over UDP. As we explained in the footnote to the Abry-Veitch method, the octave j in
the logscale diagram also expresses the timescale of the aggregated process. Based on
this insight it is possible to determine over which timescales scaling occurs. We start
with an initial bin size of 100 ms, which leads to values of t
j
=0.1, 0.2, 0.4, 0.8, 1.6,
3.2, ... for j=1, 2, 3, 4, 5, ... , respectively. For all processes in which we observed a
scaling like the one depicted in Figure 5 and Figure 7, the knee-point at which the lin-
ear scaling starts is at about j=5 or j=6, i.e., respectively 1.6 seconds or 3.2 seconds.
In [16] the authors show that the average download time per WAP page, including
embedded objects, is in the order of 1.5-3 seconds. Hence, this timescale marks the
demarcation line between the WAP transport layer protocol and the user behavior.
2
4
6
8
10
12
14
22
24
26
28
30
32
Octave j
y
j
Fig. 7. Typical plot for processes showing
long-range dependency. Below a certain
scale no regular linear scaling exists
2
4
6
8
10
12
14
6
8
10
12
14
16
18
Octave j
y
j
Fig. 8. Linear scaling over the whole range.
Although there is a step at j=9, the slope on
both sides is almost the same. This indicates
second- order self-similarity
6. Conclusion
Based on live GPRS measurements, we applied 6 different established Hurst esti-
mation methods, including the comprehensive and robust Abry-Veitch test on packet
arrival and data volume processes. We showed the Hurst value as well as the scaling
behavior for the busy-hour for 3 different traffic types and two different networks. In
the case of aggregated traffic and also in the case of individual WAP and WEB traffic
traces, the results strongly suggest long-range dependency.
We confirmed that the dominant traffic type (WAP or WEB) determines the degree
of self-similarity of the aggregated traffic. In our case however, the minor traffic al-
ways exhibits a Hurst value larger than the Hurst value of the major traffic. This is
particularly interesting as WAP traffic is based on UDP.
We showed that WAP traffic has a very different scaling behavior compared to
WEB traffic for small scales. We identified the demarcation line between small and
larger scales for WAP to coincide with the average page duration. Larger scales can
probably be accounted to user behavior.
As future research we wish to investigate the reason(s) why the minor traffic frac-
tion exhibits higher Hurst parameter values for all tests.
Further research also includes investigating the reason(s) of self-similarity of WAP
traffic and the exact nature of scaling; whether it is mulitfractional, monofractional,
strictly self-similar or `just' long-range dependent.
Acknowledgements
We thank Vodafone for providing us with live GPRS traces from various networks
in Europe. We also thank Thomas Karagiannis, Michalis Faloutsos, the authors of the
SELFIS tool, and Darryl Veitch, the author of the LD estimator-code, for making
their software available on the Internet.
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